The purpose of this paper is to study a boundary reaction problem on the space $X \times \mathbb R$, where $X$ is an abstract
Wiener space. We prove that smooth bounded solutions enjoy a symmetry property, i.e., are one-dimensional in a suitable
sense. As a corollary of our result, we obtain a symmetry property for some solutions of the following equation
$(-\Delta_\gamma)^s u= f(u)$,
with $s\in (0,1)$, where $(-\Delta_\gamma)^s$ denotes a fractional power
of the Ornstein-Uhlenbeck operator, and we prove that for any $s \in (0,1)$
monotone solutions are one-dimensional.